gpt pro 5.2 attempt on Math-961

FORMAL RESTATEMENT.

Fix $t\in\mathbb N$ with $t\ge 1$. For each $n\in\mathbb N$, let $\mathcal D(n)$ be the set of positive divisors of $n$. Say that $n$ is $t$-representable if there exists a finite subset $S\subseteq \mathcal D(n)$ of distinct divisors such that \[ \sum_{d\in S} d = t. \] Let \[ S_t := \{n\in\mathbb N : n\text{ is $t$-representable}\}. \] If the natural density \[ d_t := \lim_{X\to\infty} \frac{|S_t\cap\{1,2,\dots,X\}|}{X} \] exists, the problem asks whether there are constants $c_1,c_2>0$ such that \[ d_t \sim \frac{c_1}{(\log t)^{c_2}} \qquad (t\to\infty). \]

QUICK LITERATURE/CONTEXT CHECK.

The problem statement records that Erdős proved $d_t$ exists and that there are positive constants $c_3,c_4$ with \[ (\log t)^{-c_3} < d_t < (\log t)^{-c_4}. \] We do not assume any more literature.

ATTACK PLAN.

We prove two unconditional structural facts:

$1$ For each fixed $t$, the membership of $n$ in $S_t$ depends only on divisibility of $n$ by integers $\le t$, hence is periodic modulo $\mathrm{lcm}(1,2,\dots,t)$. This immediately implies $d_t$ exists (and is rational).

$2$ Provide elementary lower bounds on $d_t$ (e.g. $d_t\ge 1/t$ via multiples of $t$).

We then compute $d_t$ exactly for small $t$ as a sanity check.

WORK.

Lemma 859.1 (Periodicity and existence of $d_t$).

Lemma. Fix $t\ge 1$ and let $L:=\mathrm{lcm}(1,2,\dots,t)$. Then $S_t$ is a union of residue classes modulo $L$. In particular, the natural density $d_t$ exists and equals \[ d_t = \frac{1}{L}\,\#\{0\le r<L : r\in S_t\}. \]

Proof. Let $n\in\mathbb N$. If $n$ is $t$-representable, then there exists a subset $S\subseteq\mathcal D(n)$ with $\sum_{d\in S} d=t$. Every $d\in S$ satisfies $d\le t$ because all summands are positive. Thus the condition "$n$ is $t$-representable" depends only on which integers $d\in\{1,2,\dots,t\}$ divide $n$.

For each $d\le t$, the predicate "$d\mid n$" depends only on the residue class of $n$ modulo $d$, hence only on $n\bmod L$ because $d\mid L$. Therefore the set of residues $r\bmod L$ for which the collection of divisors $\{d\le t: d\mid r\}$ admits a subset summing to $t$ is well-defined; call this set $R\subseteq\{0,1,\dots,L-1\}$. Then $n\in S_t$ if and only if $n\bmod L\in R$, i.e. $S_t$ is a union of residue classes mod $L$.

Any union of residue classes modulo $L$ has natural density $|R|/L$ by periodicity: among $1,2,\dots,X$ each residue appears $\lfloor X/L\rfloor$ or $\lceil X/L\rceil$ times, and the proportion tends to $|R|/L$ as $X\to\infty$. Thus $d_t$ exists and equals $|R|/L$. $\square$

Lemma 859.2 (A trivial universal lower bound).

Lemma. For every $t\ge 1$, we have $d_t\ge 1/t$.

Proof. If $n$ is a multiple of $t$, then $t\in\mathcal D(n)$, and we can represent $t$ as the sum of the single divisor $t$. Hence every multiple of $t$ lies in $S_t$. The set of multiples of $t$ has natural density $1/t$. Therefore $d_t\ge 1/t$. $\square$

Lemma 859.3 (A union-bound upper estimate in terms of distinct partitions).

Lemma. Let $\mathcal P_t$ be the set of all finite subsets $S\subseteq\{1,2,\dots,t\}$ with $\sum_{d\in S} d=t$, and for $S\in\mathcal P_t$ let $m(S):=\mathrm{lcm}(S)$. Then \[ d_t \le \sum_{S\in\mathcal P_t} \frac{1}{m(S)}. \]

Proof. If $n\in S_t$, then there exists $S\in\mathcal P_t$ such that every $d\in S$ divides $n$, hence $m(S)=\mathrm{lcm}(S)$ divides $n$. Therefore \[ S_t \subseteq \bigcup_{S\in\mathcal P_t} \{n: m(S)\mid n\}. \] Taking natural densities and using the union bound for densities of sets, \[ d_t \le \sum_{S\in\mathcal P_t} \mathrm{dens}(\{n: m(S)\mid n\}) = \sum_{S\in\mathcal P_t} \frac{1}{m(S)}. \] $\square$

Fast reality check (exact values for small $t$).

Using Lemma 859.1 (period $L=\mathrm{lcm}(1,\dots,t)$) and a direct computation for $t\le 10$, we obtain the exact values: \[ \begin{array}{c|c|c} t & d_t & \text{decimal}\\\hline 1 & 1 & 1.0000\\ 2 & 1/2 & 0.5000\\ 3 & 2/3 & 0.6667\\ 4 & 1/2 & 0.5000\\ 5 & 7/15 & 0.4667\\ 6 & 7/15 & 0.4667\\ 7 & 16/35 & 0.4571\\ 8 & 3/7 & 0.4286\\ 9 & 119/315 & 0.3778\\ 10 & 2/5 & 0.4000 \end{array} \]

VERIFICATION.

• Lemma 859.1: the key point is that any divisor used in a sum to $t$ is automatically $\le t$, ensuring finiteness and periodicity modulo $\mathrm{lcm}(1,\dots,t)$.
• Lemma 859.2 is valid because using the single divisor $t$ is allowed ("distinct divisors").
• The computed small values agree with obvious checks: $d_1=1$ and $d_2=1/2$ (only even $n$ have divisor $2$).

FINAL. UNRESOLVED.

(i) Strongest proved partial results. We proved that for every fixed $t$, the set $S_t$ is periodic modulo $\mathrm{lcm}(1,\dots,t)$, hence $d_t$ exists and is rational. We also proved the universal lower bound $d_t\ge 1/t$ and an explicit union-bound upper estimate in Lemma 859.3. Exact values for $t\le 10$ were computed.

(ii) First gap. Determine the true asymptotic behaviour of $d_t$ as $t\to\infty$; in particular, prove or disprove the existence of $c_1,c_2>0$ with $d_t\sim c_1/(\log t)^{c_2}$.

(iii) Top 3 next moves. $1$ Analyse which subsets $S\subseteq\{1,\dots,t\}$ with sum $t$ give the smallest $\mathrm{lcm}(S)$; these dominate the union-bound in Lemma 859.3. $2$ Develop a lower-bound construction beyond $1/t$ by forcing a "binary" divisor structure (e.g. many powers of two) with reasonably high density. $3$ Compute $d_t$ for larger $t$ (say $t=20,30,40$) using periodicity but avoiding enumerating all residues by instead enumerating the moduli $m(S)$ and using inclusion--exclusion on their lcms.

(iv) What a minimal counterexample would look like. If $d_t$ fails to be a power of $\log t$, then either (a) $d_t$ is much larger, requiring unexpectedly many small-lcm partitions of $t$ into distinct parts, or (b) $d_t$ is much smaller, meaning that every representation of $t$ as a sum of distinct integers forces an lcm so large that the union of multiples has unusually small density.